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Cell structures and completely metrizable spaces and their mappings. (English) Zbl 1382.54018

The authors study completely metrizable spaces from a geometric point of view, via cell structures. They define an inverse sequence of graphs \(\{(G_i,g^j_{i})\}\) from a family of graphs \(\{(G_i,r_i)\}_{i\in\mathbb N}\), and \(\{g^j_i\}_{j\geq i}\) a family of continuous functions \(g^j_i:G_j\longrightarrow G_i\) satisfying certain axioms. This gives a cell structure \[ G_0 \stackrel{g^1_0}{\longleftarrow} G_1 \stackrel{g^2_1}{\longleftarrow} G_3 \longleftarrow \cdots \] and the inverse limit of the inverse sequence \(G_\infty\) (the set of all threads of the topological product \(\prod_{i\in\mathbb{N}} G_i\), where each \(G_i\) has the discrete topology). Under some additional conditions on the graphs \((G_i,r_i)\), \(i\in\mathbb N\), an equivalence relation on \(G_\infty\) is obtained given by \(x\sim y\) iff \((x(i),y(i))\in r_i\) for each \(i\in\mathbb N\). If \(\pi:G_\infty\to G^\ast\) is the natural quotient map and \(G^\ast=G_\ast/\sim\) is endowed with the quotient topology, the authors prove that for each cell structure, \(\pi\) is a perfect mapping and \(G^\ast\) is a completely metrizable space (Theorem 3.6). Also they show (Theorem 4.3) that every completely metrizable space can be obtained from a complete cell structure like in Theorem 3.6. These results are applied in the case of the real line, Euclidian spaces, Hilbert spaces and Polish spaces. Using cell maps between complete cell structures, a continuous function between the spaces determined by cell structures is obtained, and every function between completely metrizable spaces comes from such a cell map.

MSC:

54E50 Complete metric spaces
54C05 Continuous maps
54B35 Spectra in general topology
03D78 Computation over the reals, computable analysis
78A70 Biological applications of optics and electromagnetic theory
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